8. Power Series

Inside the radius of convergence — Abel and Tauber theorems, rearrangements, Cauchy products.

• Power Series and Radius of Convergence — Where does a power series converge, and where does it fall apart? Inside the radius lies a safe zone. There the series converges even with absolute values. Outside, the terms themselves fail to shrink to zero. Points on the boundary each have their own fate. For example, think of Wi-Fi in a 🏠 house. It works well inside and drops as you walk away. Right at the door it may or may not connect, so you check case by case.
• Hadamard Gap Theorem — What happens at the boundary when a series has growing gaps? Exponents doubling like 1, 2, 4, 8 leave bigger and bigger holes. The holes block continuation to every point of the circle. This happens even though the series converges nicely inside. The gap-free comparison series stays calm. For example, imagine a 🪜 staircase whose steps spread by 1, 2, 4, 8. The first steps are easy. Soon one step is too far, and the wall's end is out of reach.
• Abel's Theorem — Does the sum of a series match the limit of its power series at 1? If the series converges, the power series heads to the same value as x nears 1, as in the formula below. Even the divergent Grandi series points to one half in this limit. But that value is an Abel sum, a separate convention, not the sum. For example, imagine a 🧑 friend shouting 1 and 0 in turns. If each shout is a bit quieter than the last, the running average settles near one half.
• Tauber's Theorem — If an Abel sum exists, must the ordinary sum exist too? Only under an extra condition. The terms must shrink faster than one over n. Grandi's series breaks it, since its terms never shrink. When the condition holds, the Abel-sum curve reaches the ordinary sum. For example, think of walking to the door of a 🏠 house. Your steps must keep shrinking to stop right at the door. Steps that never shrink just carry you past it, back and forth.
• Two Roads to e — Two roads lead to e — which one gets there faster? The compound-growth road creeps forward step by step. The factorial series has terms that shrink almost instantly. A handful of terms already pins down many digits. For example, think of 💰 money in a 🐷 piggy bank earning interest on its interest. More rounds push it toward e, but the series arrives far sooner.
• ζ(4) and Even Powers — What do the reciprocals of fourth powers add up to? The partial sums hug the answer after only a few terms. Fourth powers shrink the tail to almost nothing. At every even power, the closed answer is a power of pi times a fraction. For example, picture stacking 🍰 cake slices that shrink fast. Each slice is soon tiny, so a few slices fix the total height.
• Rearranging Changes the Sum — What happens to the sum when we reorder a conditionally convergent series? Positive terms and negative terms both pile up without end. So choosing which side to draw from first pushes the sum around. Nothing like this happens when the absolute series converges. For example, think of putting 💰 coins into a 🐷 piggy bank and taking them out. If both piles of coins are endless, the order alone changes the balance.
• Absolute Convergence — If the absolute-value series converges, must the original follow? A finite absolute total settles everything. The signed partial sums wobble inside it and land on one point. In reverse, alternating signs can rescue a series whose absolute total diverges. For example, picture a 🧑 person stepping forward, then a little back. If all the step lengths add to a finite total, the walker ends up standing somewhere definite.
• Riemann's Rearrangement Theorem — Can rearranging alone manufacture any sum we want? The greedy rule is simple. Below the target, take a positive term; above it, take a negative one. The terms shrink, so each overshoot shrinks too. That is why any real target can be hit exactly. For example, think of steering a 🚗 car. Turn more when short of the spot, back when past it. Small corrections land you exactly where you aim.
• The Cauchy Product — How do the terms pair up when two series are multiplied? Each product term gathers pairs along a diagonal of the lattice, as in the formula below. If one side converges absolutely, the product lands on the product of the sums. If both converge only conditionally, the diagonal sums can collapse. For example, picture 📦 boxes in a grid, picked up one diagonal at a time. If the boxes shrink fast enough, the basket's total settles down.